Giancarlo Mauceri

Higher-order Riesz operators for the Ornstein-Uhlenbeck semigroup

We prove that the second-order Riesz transforms associated to the Ornstein–Uhlenbeck semigroup are weak type (1,1) with respect to the Gaussian measure in finite dimension. We also show that they are given by a principal value integral plus a constant multiple of the identity. For the Riesz transforms of order three or higher, we present a counterexample showing that the weak type (1,1) estimate fails.

Spectral multipliers for the Ornstein-Uhlenbeck semigroup

The setting of this paper is Euclidean space with the Gaussian measure. We letL be the associated Laplacian, by means of which the Ornstein-Uhlenbeck semigroup is defined. The main result is a multiplier theorem, saying that a function ofL which is of Laplace transform type defines an operator of weak type (1,1) for the Gaussian measure. The (distribution) kernel of this operator is determined, in terms of an integral involving the kernel of the Ornstein-Uhlenbeck semigroup. This applies in particular to the imaginary powers ofL. It is also verified that the weak type constant of these powers increases exponentially with the absolute value of the exponent.

Hp multipliers on stratified groups

SummaryIn this paper we give a criterion for boundedness on the Hardy spaces for functions M(ℒ) of the sublaplacian ℒ on a stratified group. The criterion requires that the function M satisfies locally a Besov condition. The proof is based on the atomic and molecular characterization of Hardy spaces.

Damping oscillatory integrals

Let S be a smooth compact convex hypersurface of finite type in R n+l, with surface measure #, and let u be a nonnegative-real-valued function on S. We shall study the behaviour of the Fourier transform of the measure u#. In particular, we are interested in obtaining inequalities for the decay of (u#) (4) as 4 goes to inifinty. This problem has been considered by various authors, including Hlawka [5], Herz [4], Littman [10], Randol [12], and Svensson [15]. For an overview of such results and their significance in harmonic analysis, we refer our readers to Stein [14]. By taking a partition of unity, and using suitable coordinate systems, the study of the Fourier transform integral may be reduced to that of an oscillatory integral I of the form 1(2)= ~ v(x)exp(i2(a(x))dx V2~R, R.

The Weyl transform and bounded operators on Lp(Rn)

A multiplier theorem for the Weyl transform is proved. This theorem is used to derive sufficient conditions for the boundedness of a general operator on Lp(Rn). An application to multipliers of the Hermite expansion is given.

Maximal operators for the holomorphic Ornstein-Uhlenbeck semigroup

For each p in [1, ∞) let Ep denote the closure of the region of holomorphy of the Ornstein-Uhlenbeck semigroup {Ht : t> 0} on L p with respect to the Gaussian measure. We prove sharp weak type and strong type estimates for the maximal operator f �→ H ∗ f = sup{|Hzf | : z ∈ Ep} and for a class of related operators. As a consequence of our methods, we give a new and simpler proof of the weak type 1 estimate for the maximal operator associated to the Mehler kernel.

A hardy space associated with twisted convolution

Twisted convolution has been investigated in connection with the Weyl functional calculus by several authors [ 1, 6, 141. More recently, LP-boundedness of twisted convolution operators has been studied by one of us [lo], who proved a multiplier theorem of Hormander type. The purpose of this paper is to define and describe a Hardy space 3” naturally associated with twisted convolution operators. In order to define this space, we consider certain maximal operators in terms of twisted convolutions. The Hardy space will then consist of those distributions which are mapped into L’(V) by any one of these maximal operators. Some of the maximal operators we consider are the analogues in the twisted case of those which define H’ in the theory of Fefferman and Stein

Sharp estimates for the Ornstein-Uhlenbeck operator

Let L be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure γ on Rd. We prove a sharp estimate of the operator norm of the imaginary powers of L on Lp(γ), 1 < p < ∞. Then we use this estimate to prove that if b is in [0,∞) and M is a bounded holomorphic function in the sector {z ∈ C : |arg(z − b)| < arcsin |2/p−1|} and satisfies a Hormander-like condition of (nonintegral) order greater than one on the boundary, then the operator M(L) is bounded on Lp(γ). This improves earlier results of the authors with J. Garcia-Cuerva and J.L. Torrea.

Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator..

In the setting of Euclidean space with the Gaussian measure γ , we consider all first-order Riesz transforms associated to the infinitesimal generator of the Ornstein–Uhlenbeck semigroup. These operators are known to be bounded on L p (γ) , for 1<p<∞ . We determine which of them are bounded from H 1 (γ) to L 1 (γ) and from L ∞ (v) to BMO(γ ). Here H 1 (γ) and BMO(γ ) are the spaces introduced in this setting by the first two authors. Surprisingly, we find that the results depend on the dimension of the ambient space.

On maximal functions

SuntoSi dà una nuova dimostrazione, più semplice, del teorema di E. M. Stein sulla funzione massimale sferica; si trova qualche generalizzazione.SummaryWe give a new, simpler, version of E. M. Stein’s theorem on the spherical maximal function, and offer a generalisation.